Existence and multiplicity of nontrivial solutions for quasilinear elliptic systems

The existence and multiplicity of nontrivial solutions are obtained for the quasilinear elliptic systems by the linking argument, the cohomological index theory and the pseudo-index theory.     

Existence and nonexistence of nontrivial weak solution for a class of general capillarity systems

The aim of this paper is to study the nonexistence and existence of nonnegative, nontrivial weak solution for a class of general capillarity systems. The proofs rely essentially on the minimum principle combined with the mountain pass theorem.     

Existence of a nontrivial weak solution to quasilinear elliptic equations with singular weights and multiple critical exponents

In this paper, we consider the existence of a non-trivial weak solution to a quasilinear elliptic equation with singular weights and multiple critical exponents in the whole space. Firstly, we get the existence of a local Palais–Smale sequence by verifying the geometric conditions of the Mountain Pass Lemma. Secondly, we study the concentration properties of the Palais–Smale sequence of a zero weak limit. Thirdly, we deduce by contradiction to eliminate the possibility of a zero weak limit case. Lastly, applying a monotonic inequality, we shall prove that the nontrivial weak limit of the Palais–Smale sequence is indeed a weak solution.     

Existence and nonexistence results for critical growth polyharmonic elliptic systems

In this work, we consider semilinear elliptic systems for the polyharmonic operator having a critical growth nonlinearity. We establish conditions for existence and nonexistence of nontrivial solutions to these systems.     

Existence and uniqueness of solutions for quasilinear elliptic systems

We obtain necessary and sufficient conditions for the existence of positive solutions for a class of sublinear Dirichlet quasilinear elliptic systems.

     

Existence and multiplicity results for quasilinear elliptic differential systems

We use a nonsmooth critical point theory to prove existence results for a variational system of quasilinear elliptic equations in both the sublinear and superlinear cases. We extend a technique of Bartsch to obtain multiplicity results when the system is invariant under the action of a compact Lie group. The problem is rather different from its scalar version, because a suitable condition on the coefficients of the system seems to be necessary in order to prove the convergence of the Palais-Smale sequences. Such condition is in some sense a restriction to the "distance" between the quasilinear operator and a semilinear one.     

Multiplicity of nontrivial solutions for quasilinear elliptic equation

A three-critical-point theorem related to local linking is obtained and applied to study multiplicity of nontrivial solutions for a class of quasilinear elliptic equations.     

Existence of a nontrivial solution for a class of superquadratic elliptic problems

A nontrivial solution is obtained for a class of superquadratic elliptic problems by variational methods.     

Existence results for a class of degenerate quasilinear elliptic systems

We establish existence results for weak solutions of degenerate quasilinear elliptic systems. By using the variational method we obtain the existence of a solution for an elliptic system with Dirichlet boundary condition under some restriction on λ.     

Existence and nonexistence of ground state solutions for elliptic equations with a convection term

We deal with the existence of positive solutions u decaying to zero at infinity, for a class of equations of Lane-Emden-Fowler type involving a gradient term. One of the main points is that the differential equation contains a semilinear term σ(u) where σ:(0,∞)→(0,∞) is a smooth function which can be both unbounded at infinity and singular at zero. Our technique explores symmetry arguments as well as lower and upper solutions.     

Existence of three nontrivial solutions for elliptic systems with critical exponents and weights

In this paper, an elliptic system with critical Sobolev exponents and weights is studied by the linking theorem on product space and the classical minimax theorem. By investigating the coefficients of the critical nonlinearities, we establish the existence of three nontrivial solutions.     

Existence and nonexistence of solutions for singular quadratic quasilinear equations

We study both existence and nonexistence of nonnegative solutions for nonlinear elliptic problems with singular lower order terms that have natural growth with respect to the gradient, whose model is

     

Existence of a unique solution to a quasilinear elliptic equation

The purpose of this paper is to prove the existence of a unique classical solution u(x) to the quasilinear elliptic equation −∇⋅(a(u)∇u)+v⋅∇u=f, where u(x₀)=u₀ at x₀∈Ω and where n⋅∇u=g on the boundary ∂Ω. We prove that if the functions a, f, v, g satisfy certain conditions, then a unique classical solution u(x) exists. Applications include stationary heat/diffusion problems with convection and with a source/sink, where the value of the solution is known at a spatial location x₀∈Ω, and where n⋅∇u is known on the boundary.     

带有变指数拟线性椭圆方程组的边界爆破解

研究了在光滑有界域中带有变指数的拟线性椭圆方程组,且该方程组满足边界爆破的条件,在常指数的基础上进一步深入讨论了变指数的情况.主要运用了构造上下解和迭代的方法证明了边界爆破解在临界与次临界条件下,解的存在性,唯一性以及边界行为.     

Multiple symmetric results for singular quasilinear elliptic systems with critical homogeneous nonlinearity

This paper deals with the existence and multiplicity of symmetric solutions for a class of singular quasilinear elliptic systems with critical homogeneous nonlinearity in a bounded symmetric domain. Applying variational methods and the symmetric criticality principle of Palais, we establish several existence and multiplicity results of G‐symmetric solutions under some appropriate assumptions on the weighted functions and the parameters. Copyright © 2016 John Wiley & Sons, Ltd.     

Existence of three nontrivial solutions for an elliptic system

In this paper we consider the existence of nontrivial solutions for an elliptic system, where the nonlinear term is superlinear in one equation and sublinear in the other equation. By constructing two cones and computing the fixed point index in K₁, K₂ and K₁×K₂, we obtain that the elliptic system has three nontrivial solutions (u,0), (0,v) and (u^∗,v^∗). It is remarkable that the third nontrivial solution (u^∗,v^∗) is established on the Cartesian product of two cones, in which the feature of two equations can be exploited better.     

Existence of solutions for degenerate quasilinear elliptic equations

This paper deals with a class of degenerate quasilinear elliptic equations of the form −div(a(x,u,∇u)=g−div(f), where a(x,u,∇u) is allowed to be degenerate with the unknown u. We prove existence of bounded solutions under some hypothesis on f and g. Moreover we prove that there exists a renormalized solution in the case where g∈L¹(Ω) and f∈(L^p′(Ω))^N.